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Many of the known functions are continuous in open interval. Polynomial, trigonometric, exponential, logarithmic functions etc. are continuous functions in open interval.
The possibility that there always exist a point around a given point is not there at end points of closed interval. We can not determine left limit at lower end and right limit at upper end of the closed interval. For this reason, we test continuity of function at the closing points from one side only. For a function to be continuous in the closed interval, it should be continuous at all points in the interval and also at the bounding values of closed intereval, [a,b]. Hence,
(i) limit exists at all points in the interval and are equal to function values at those points.
$$\underset{x>c}{\overset{}{\mathrm{lim}}}f\left(x\right)=f\left(c\right);a<c<b$$
(ii) right limit exists at x=a and is equal to function value at the lower end of closed interval. .
$$\underset{x>a+}{\overset{}{\mathrm{lim}}}f\left(x\right)=f\left(a\right)$$
(iii) left limit exists at x=b and is equal to function value at the upper end of closed interval.
$$\underset{x>b-}{\overset{}{\mathrm{lim}}}f\left(x\right)=f\left(a\right)$$
If two functions are continuous at a point or in interval, then function resulting from function operations like addition, subtraction, scalar product, product and quotient are continuous at that point. Further, properly formed function compositions of two or more functions are also continuous.
These properties of continuity are extremely helpful tool for determining continuity of more complicated functions, which are formed from basic functions. Idea is that we are aware of continuity of basic functions. Therefore, continuity of functions formed from these basic functions will also be continuous.
Generally, basic functions are continuous in real numbers set R or its subsets. For example, we know that polynomial functions, sine, cosine, tangent, exponential and logarithmic functions etc are continuous on R. Similarly, a radical function is continuous for non-negative x values. Their composition or the new function will be continuous in the new domain, which is defined in accordance with the rule given here :
In the nutshell, the function formed from other functions is continuous in new domain as defined above. If we look closely at the definition of continuity here, then "finding interval in which function is continuous" is same as finding "domain" of new function arising from mathematical operations.
Many functions are not defined at singularities. For example, rational functions are not defined for values of x when denominator becomes zero. By including these singular points or exception points in the domain, we can redefine function such that it becomes continuous in the extended domain. This extension of the domain of function such that function remains continuous is known as continuous extension.
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